A322817 a(n) = A001222(A065642(n)) - A001222(n), where A065642(n) gives the next larger m that has same prime factors as n (ignoring multiplicity), and A001222 gives the number of prime factors, when counted with multiplicity.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 0, 1, 1, -1, 1, 2, 1, 1, 1, 1, 1, -1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 2, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Author

Antti Karttunen, Dec 27 2018

Keywords

• sign

Examples

			For n = 2 = 2^1, the next larger number with only 2's as its prime factors is 4 = 2^2, thus a(2) = 1.
For n = 12 = 2^2 * 3^1, the next larger number with the same prime factors is 18 = 2^1 * 3^2, with the same value of A001222, thus a(12) = 0.
For n = 40 = 2^3 * 5^1, the next larger number with the same prime factors is 50 = 2^1 * 5^2. While 40 has 3+1 = 4 prime factors in total, 50 has 1+2 = 3, thus a(40) = 3 - 4 = -1.
For n = 50, the next larger number with the same prime factors is 80 = 2^4 * 5^1, thus a(50) = (4+1)-(2+1) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A001222, A065642, A322818.

Programs

PARI

A007947(n) = factorback(factorint(n)[, 1]);
A065642(n) = { my(r=A007947(n)); if(1==n, n, n = n+r; while(A007947(n) <> r, n = n+r); n); };
A322817(n) = (bigomega(A065642(n)) - bigomega(n));

Formula

a(n) = A001222(A065642(n)) - A001222(n).